Obtaining the statistically best estimate of an unobserved stochastic process based on a stochastic observation process generated by it is known as the optimal filtering problem [1]}, [2]}, [3]}, [4]}. The optimal filtering problem is equivalent to determining the evolution of the conditional probability density of the state of the unobserved process given the observation process up to the current time. When the stochastic processes are modeled as Itô type of stochastic differentiation equations (SDEs), under suitable regularity conditions, the evolution of the conditional density is described by a stochastic partial differential equation (SPDE) known as the Kushner–Stratonovich equation [1]}, [6]}, [7]}, [8]}, [9]}. The Kushner–Stratonovich equation is a nonlinear SPDE that has a complicated structure which makes it hard to solve [10]}. [11]} introduced an alternative representation to the Kushner–Stratonovich equation, resulting in a linear SPDE of an unnormalized conditional density of the state. The solutions of the Zakai equation are, nevertheless, still hard to obtain as they are generally infinite-dimensional except when the Lie algebra naturally associated with it is finite-dimensional [12]}, [13]}. In most cases, we can only hope for approximations of the solutions.
